Browsing by Author "McCoy, Robert A."
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- A Classification of some Quadratic AlgebrasMcGilvray, H. C. Jr. (Virginia Tech, 1998-07-02)In this paper, for a select group of quadratic algebras, we investigate restrictions necessary on the generators of the ideal for the resulting algebra to be Koszul. Techniques include the use of Gröbner bases and development of Koszul resolutions. When the quadratic algebra is Koszul, we provide the associated linear resolution of the field. When not Koszul, we describe the maps of the resolution up to the instance of nonlinearity.
- Closability of differential operators and subjordan operatorsFanney, Thomas R. (Virginia Polytechnic Institute and State University, 1989)A (bounded linear) operator J on a Hilbert space is said to be jordan if J = S + N where S = S* and N² = 0. The operator T is subjordan if T is the restriction of a jordan operator to an invariant subspace, and pure subjordan if no nonzero restriction of T to an invariant subspace is jordan. The main operator theoretic result of the paper is that a compact subset of the real line is the spectrum of some pure subjordan operator if and only if it is the closure of its interior. The result depends on understanding when the operator D = θ + d/dx : L²(μ) —> L²(v) is closable. Here θ is an L²(μ) function, μ and v are two finite regular Borel measures with compact support on the real line, and the domain of D is taken to be the polynomials. Approximation questions more general than what is needed for the operator theory result are also discussed. Specifically, an explicit characterization of the closure of the graph of D for a large class of (θ, μ, v) is obtained, and the closure of the graph of D in other topologies is analyzed. More general results concerning spectral synthesis in a certain class of Banach algebras and extensions to the complex domain are also indicated.
- The cohomology rings of classical Brauer tree algebrasChasen, Lee A. (Virginia Tech, 1995-07-03)In this dissertation a simple algorithm is given for calculating minimal projective resolutions of nonprojective indecomposable modules over Brauer tree algebras. Those calculated resolutions lead to an algorithm for calculating a minimal set of generators for the cohomology ring of a Brauer tree algebra.
- The compact-open topology on C(X)Ntantu, Ibula (Virginia Polytechnic Institute and State University, 1985)This paper investigates the compact-open topology on the set of Ck(X) of continuous real-valued functions defined on a Tychonoff space X. More precisely, we study the following problem: If P is a topological property, does there exist a topological property Q so that Ck(X) has P if and only if X has Q? Characterizations of many properties are obtained throughout the thesis, sometimes modulo some “mild” restrictions on the space X. The main properties involved are summarized in a diagram in the introduction.
- Complete function spacesMcCoy, Robert A. (Hindawi, 1983-01-01)A study is made of certain completeness properties of the space of allcontinuous real-valued functions on a space, where this function space has the compact-open topology.
- Covering properties and quasi-uniformities of topological spacesJunnila, Heikki J. K. (Virginia Tech, 1978-06-05)This thesis deals with the relationships between covering properties and properties of compatible quasi-uniformities of a topological space. The covering properties considered in this work are orthocompactness, metacompactness and paracompactness; some generalizations of orthocompactness are also defined and studied.
- Discrete Riemann Maps and the Parabolicity of TilingsRepp, Andrew S. (Virginia Tech, 1998-05-04)The classical Riemann Mapping Theorem has many discrete analogues. One of these, the Finite Riemann Mapping Theorem of Cannon, Floyd, Parry, and others, describes finite tilings of quadrilaterals and annuli. It relates to several combinatorial moduli, similar in nature to the classical modulus. The first chapter surveys some of these discrete analogues. The next chapter considers appropriate extensions to infinite tilings of half-open quadrilaterals and annuli. In this chapter we prove some results about combinatorial moduli for such tilings. The final chapter considers triangulations of open topological disks. It has been shown that one can classify such triangulations as either parabolic or hyperbolic, depending on whether an associated combinatorial modulus is infinite or finite. We obtain a criterion for parabolicity in terms of the degrees of vertices that lie within a specified distance of a given base vertex.
- Examples and theorems for generalized paracompact topological spacesFast, Stephen Hardin (Virginia Tech, 1990-12-05)In this thesis we answer a number of unsolved problems in generalized paracompact topological spaces. Examples satisfying the T₄ separation axiom are constructed showing the relationship between the properties B(D, ω₀)-refinability, B(D, λ)-refinability, and weak θ̅-refinability. The properties B(D, λ)-refinability and weak θ̅-refinability are shown to be strictly weaker than B(D, ω₀)-refinability. Sum theorems, mapping theorems, and o—product theorems are obtained for B(D, ω₀)-refinability, weak θ̅-refinability, and several other properties. The σ—product theorem for B(D,ω₀)-refinability, weak θ̅-refinability, and other properties are shown to follow from a new special B(D,ω₀) sum theorem.
- The fine topology and other topologies on C(X,Y)Eklund, Anthony D. (Virginia Tech, 1978-05-05)"The Fine Topology" C(X,Y) where (Y,d) is a metric space is referred to, in an exercise in [14], as the topology generated by basic open neighborhoods of the form B(f,E) = {g: d(f(x),g(x)) < E(x)} where E is a positive continuous real valued function. So in the fine topology, a function g is close to f if g(x) is continuously close to f(x); whereas in the uniform topology, g(x) must be uniformly close to f(x), that is, within a constant distance of f(x). So the fine topology is an obvious refinement of the uniform topology. This topology has not been extensively studied before, and it is the purpose of this paper to see how the fine topology fits in with the lattice of other well studied topologies on C(X,Y), and to study some properties of this topology in itself. Furthermore, other results on these well studied topologies will-be examined and compared with the fine topology.
- Fine topology on function spacesMcCoy, Robert A. (Hindawi, 1986-01-01)This paper studies the topological properties of two kinds of fine topologies on the space C(X,Y) of all continuous functions from X into Y.
- k-space function spacesMcCoy, Robert A. (Hindawi, 1980-01-01)A study is made of the properties on X which characterize when Cπ(X) is a k-space, where Cπ(X) is the space of real-valued continuous functions on X having the topology of pointwise convergence. Other properties related to the k-space property are also considered.
- Lie derivations on rings of differential operatorsChung, Myungsuk (Virginia Tech, 1995-04-04)Derivations on rings of differential operators are studied. In particular, we ask whether Lie derivations are forced to be associative derivations. This is established for the Weyl algebras, which provides the details of a theorem of A. Joseph. The ideas are extended to localizations of Weyl algebras. As a corollary, the implication is verified for the universal enveloping algebras of nilpotent Lie algebras.
- One-to-one correspondance between maximal sets of antisymmetry and maximal projections of antisymmetryHuang, Jiann-Shiuh (Virginia Tech, 1991-06-14)Let X be a compact Hausdorff space and A a uniform algebra on X. Let if be an isometric unital representation that maps A into bounded linear operators on a Hilbert space. This research investigated that there is a one-to-one correspondence between the collection of maximal sets of antisymmetry for A and that of maximal projections of antisymmetry for π (A) under the extension of π if π satisfies a certain regularity property.
- Parameter identification in linear and nonlinear parabolic partial differential equationsZhang, Lan (Virginia Tech, 1995)The research presented in this dissertation is carried out in two parts; the first, which is the main work of this dissertation, involves development of continuous differentiability of the solution with respect to the unknown parameters. For linear parabolic partial differential equations, only mild conditions are assumed on the admissible parameter space. The nonlinear partial differential equation we consider is a generalized Burgers’ equation, for which we establish the well-posedness and the smoothness properties of the solution with respect to the parameters. In the second part, we consider parameter identification problems for these two parameter dependent systems. The identification scheme which we use here is the quasilinearization method. Based on the results in the first part of this work, we obtain existence and local convergence of the algorithm. We also present some numerical examples which demonstrate the performance of the quasilinearization scheme.
- Pole-placement with minimum effort for linear multivariable systemsAl-Muthairi, Naser F. (Virginia Polytechnic Institute and State University, 1988)This dissertation is concerned with the problem of the exact pole-placement by minimum control effort using state and output feedback for linear multivariable systems. The novelty of the design lies in obtaining a direct transformation of the system matrices into a modified controllable canonical form. Two realizations are identified, and the algorithms to obtain them are derived. In both cases, the transformation matrix has some degrees of freedom by tuning a scalar or a set of scalars within the matrix. These degrees of freedom are utilized in the solution to reduce further the norm of the state feedback matrix. Then the pole-placement problem is solved by minimizing a certain functional, subject to a set of specified constraints. A non-canonical form approach to the problem is also proposed, where it was only necessary to transform the input matrix to a special form. The transformation matrix, in this method, has larger degrees of freedom which can be utilized in the solution. Moreover, a new pole-placement method based on the non-canonical approach is derived. The solution, in this method, was made possible by solving the Lyapunov matrix equation. Finally, an iterative algorithm for pole-placement by output feedback is extended so as to obtain an output feedback matrix with a small norm. The extension has been accomplished by applying the successive pole shifting method. Two schemes for the pole shifting are proposed. The first is to successively shift the poles through straight paths starting from the open loop poles and ending at the desired poles, whereas the second scheme shifts the poles according to a successive change of their characteristic polynomial coefficients.
- Polynomial approximation and Carleson measures on a general domain and equivalence classes of subnormal operatorsQiu, James Zhijan (Virginia Tech, 1993)This thesis consists of eight chapters. Chapter 1 contains the preliminaries: the background, notation and results needed for this work. In Chapter 2 we study the problem of when P, the set of analytic polynomials, is dense in the Hardy space Ht(G) or the Bergman space LtnG, where G is a bounded domain and t ∈ [1,∞). Characterizations of special domains are also given. In Chapter 3 we generalize the definition of a Carleson measure to an arbitrary simply connected domain. Let G be a bounded simply connected domain with harmonic measure ω. We say a positive measure τ on G is a Carleson measure if there exists a positive constant c such that for each t ∈ [1, ∞) and each polynomial p we have ⎮⎮p⎮⎮L¹(τ)≤ c ⎮⎮p⎮⎮ Lᵗ(ω), We characterize all Carleson measures on a normal domain-definition: a domain G where P is dense in H¹(G). It turns out that P is dense in Hᵗ(G) for all t when G is normal. In Chapter 4 we describe some special simply connected domains and describe how they are related to each other via various types of polynomial approximation. In Chapter 5 we study the various equivalence classes of subnormal operators under the relations of unitary equivalence, similarity and quasi similarity under the assumption that G is a normal domain. In Chapter 6 we characterize the Carleson measures on a finitely connected domain. We are able to push our techniques in the latter setting to characterize those subnormal operators similar to the shift on the closure of R(K) in L²(σ) when R(K) is a hypo dirichlet algebra. In Chapter 7 we illustrate our results by looking at their implications when G' is a crescent. Several interesting function theory problems are studied. In Chapter 8 we study arc length and harmonic measures. Let G be a Dirichlet domain with a countable number of boundary components. Let ω be the harmonic measure of G. We show that if J is a rectifiable curve and E ⊂ ∂G ∩ J is a subset with ω(E) > 0, then E has positive length.
- Problems involving relative integral bases for quartic number fieldsHymo, John A. (Virginia Tech, 1990-05-14)In this dissertation the question of whether or not a relative extension of number fields has a relative integral basis is considered. In Chapters 2 and 3 we use a criteria of Mann to determine when a cyclic quartic field or a pure quartic field has an integral basis over its quadratic subfield. In the final chapter we study the question: if the relative discriminant of an extension K / k is principal, where [K : k] = l such that l is an odd prime and k is either a quadratic or a normal quartic number field, does K / k have an integral basis?
- The property B(P,[alpha])-refinability and its relationship to generalized paracompact topological spacesPrice, Ray Hampton (Virginia Polytechnic Institute and State University, 1987)The property B(P,∝)-refinability is studied and is used to obtain new covering characterizations of paracompactness, collectionwise normality, subparacompactness, d-paracompactness, a-normality, mesocompactness, and related concepts. These new characterizations both generalize and unify many well-known results. The property B(P,∝)-refinability is strictly weaker than the property Θ-refinability. A B(P,∝)-refinement is a generalization of a σ-locally finite-closed refinement. Here ∝ is a fixed ordinal which dictates the number of "levels" in a given refinement, and P represents a property such as discreteness or local finiteness which each "level" must satisfy relative to a certain subspace.
- Rational and harmonic approximation on F.P.A. setsFerry, John (Virginia Tech, 1991-08-15)Let K be a compact subset of complex N-dimensional space, where N ≥ 1. Let H(K) denote the functions analytic in a neighborhood of K. Let R(K) denote the closure of H(K) in C(K). We study the problem: What is R(K)? The study of R(K) is contained in the field of rational approximation. In a set of lecture notes, T. Gamelin [6] has shown a certain operator to be essential to the study of rational approximation. We study properties of this operator. Now let K be a compact subset of real N-dimensional space, where N ≥ 2. Let harmK denote those functions harmonic in a neighborhood of K. Let h(K) denote the closure of harmK in C(K). We also study the problem: What is h(K)? The study of h(K) is contained in the field of harmonic approximation. As well as obtaining harmonic analogues to our results in rational approximation, we also produce a harmonic analogue to the operator studied in Gamelin's notes.
- Representation theory, Borel cross-sections, and minimal measuresMiller, Janice E. (Virginia Tech, 1993)Let E be an analytic metric space, let X be a separable metric space with a regular Borel probability measure μ and let Π: E → X be a continuous map with μ(X \ Π(E)) =0. Schwartz’s lemma states that there exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of these Borel cross-sections are in one-to-one correspondence with the representations of the form Γ:Cb(E) → L∞(μ) with Γ(f∘Π) = f for every f ∈ Cb(X). The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E. Now let E, X, and μ be as above and let Π: E → X be an onto Borel map. There exists a Borel cross-section for Π defined almost everywhere (μ). The equivalence classes of the Borel cross-sections for Π are in one-to-one correspondence with the representations of the form Γ:B(E) → L∞(μ) with Γ(f∘Π) = f for every f in Cb(X), where B(E) is the C*-algebra of the bounded Borel functions on E. The representations are also in one-to-one correspondence with equivalence classes of the minimal measures on E.